Maximal subalgebras of associative superalgebras.
The aim of the paper is the classification of maximal subalgebras of finite dimensional central simple associative superalgebras \(A=A_0\oplus A_1\) over a field \(k\). If \(Z(A)\) is the center of \(A\) then \(Z(A)_0=k\) because \(A\) is central. According to \textit{C. T. C. Wall} [J. Reine Angew. Math. 213, 187-199 (1964; Zbl 0125.01904)], there are two cases: 1) \(A\) is of even type, that is \(A\) is central simple as ungraded \(k\)-algebra, \(Z(A)_1=0\), 2) \(A\) is of odd type, that is \(A_0\) is a central simple \(k\)-algebra, \(Z(A)=k\oplus ku\), where \(u^2\in k^*\) and \(A_1=A_0u\). Suppose that \(A\) is of even type with a subalgebra \(S\) and \(V\) is an irreducible \(A\)-module, \(\Delta=\text{End}_AV\). In terms of \(V\) and special graded submodules there is given a criterion under which \(S\) is a maximal subalgebra of \(A\). Let \(A\) be of odd type. A subalgebra \(S\) in \(A\) is maximal if and only if one of the following conditions is satisfied: 3) \(S=S_0\oplus S_0u\) where \(S_0\) is a maximal subalgebra in \(A_0\), 4) \(S_0=A_0\), 5) \(A_0=C_0\oplus C_1\) is itself a graded algebra and \(S_0=C_0\oplus C_1u\). These results are extended to the case of a superalgebra \(A\) with a superinvolution *. The corresponding classification is based on results of a paper by \textit{M. Racine} [J. Algebra 30, 155-180 (1974; Zbl 0282.17009)].
scientific article; zbMATH DE number 2105786
scientific article